What happens when you flip the cover over(So the outside side is facing the inside)? In the iPad 2 there was a magnet sensor but it would not check the polarity of the magnet. In the iPad 3+(I assume the mini as well) Apple changed the sensor to check for the polarity as some people where having issues on the iPad 2, with the cover being open it would turn off when the magnets where placed to close to the sensor.

Some case manufactures where caught out by this. It could be that the case you brought has the magnets flipped to the incorrect polarity.

In reality the smart cover is not really that smart its just a bunch of magnets so the iPad should not be able to tell that your case is not an official Smart Cover.